Chapter 2.1, Problem 75E

To determine

To find:The equation that may be represented by the graph shown in figure.

Answer to Problem 75E

The given graph is represented by equations (c) and (d).

Explanation of Solution

Given data:

The set of equations is

  

  (a) $f\left(x\right)=-{\left(x-4\right)}^2+2$

  (b) $f\left(x\right)=-{\left(x+2\right)}^2+4$

  (c) $f\left(x\right)=-{\left(x+2\right)}^2-4$

  (d) $f\left(x\right)=-x^2-4x-8$

Or $f\left(x\right)=-{\left(x+2\right)}^2-4$

  (e) $f\left(x\right)=-{\left(x-2\right)}^2-4$

  (f) $f\left(x\right)=-x^2+4x-8$

Or $f\left(x\right)=-{\left(x-2\right)}^2-4$

Concept used:

The standard form of quadratic function $f\left(x\right)=a{\left(x-h\right)}^2+k$ represents a parabola with vertex $\left(h,k\right)$ . If $a<0$ then the graph of quadratic function opens downwards.

The given graph is a parabola that opens downwards so $a<0$ and vertex is in third quadrantso h and kboth are negative.

Thus, the graph is represented by the equation for which $a<0$ , $h<0$ , and $k<0$ . The quadratic function must be of the form $f\left(x\right)=-a{\left(x+m\right)}^2-n$

For equations (c) we have $a=-1$ , $h=-2$ , and $k=-4$ , and for equation (d) also $a=-1$ , $h=-2$ , and $k=-4$ .

Hence, the given graph is represented by equations (c) and (d).

Conclusion:

The given graph is represented by equations (c) and (d).